#94022

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	D₁₆
commutator	C₁₆	C₄×C₈
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1380
prime radical	C₂³	#367
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2,3,5,4,6,7,8)(9,10)(11,23)(12,29)(13,35)(14,38)(15,39)(16,22)(17,24)(18,32)(19,45)(20,30)(21,31)(25,58)(26,40)(27,61)(28,52)(33,48)(34,47)(36,65)(37,62)(41,60)(42,69)(43,57)(44,64)(46,63)(49,55)(50,53)(51,54)(56,66)(59,68)(67,71)(70,72);(9,11,13,25,36,59,12,24,31,54,64,72,30,53,45,69,16,28,37,60,14,26,34,57,15,27,32,55,63,71,33,56)(10,17,19,40,46,58,44,52,39,66,29,50,38,49,35,51,22,43,48,68,20,41,18,23,21,42,47,67,65,70,62,61)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fi,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efh,f^a=fh,h^a=hi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1218);